Sunday, September 20, 2015

DANCE DANCE TRANSVERSAL - logistics and playlists in a very crowded room (Geometry)

I originally received the very useful Dance Dance Geometry game (in PowerPoint) about six years ago from the very generous David Sladkey of Naperville High School in Naperville, Illinois. It is an amazing way to get students to practice identifying the essential angle pairs in a parallel-lines-plus-transversal situation.

A few years later, @algebrainiac (Jessica Marie) and Julie Reulbach put their own spin on it (recasting it as "Dance Dance Transversal") and gave me new ideas for how to use it.

But then... last year I arrived at my current school, with the world's tiniest classroom and 36 kids in every class. No space for everybody to move around. So no more Dance Dance Transversal for me.  *sad face*

But now... I'M BACK, BABY!

Now that Matt Vaudrey has turned me into a monster with musical cues. I've got musical cues down pat. The other day I was so pressed for time I forgot to play the theme music for my Geometry class (the opening from the old Hawaii 5-0 show), and the next day, my students said, Hey, where the heck is our theme music?!?!?

I love a self-regulating classroom.

So now I've figured out a way to do DDT even in my tiny room. Everybody gets a half-sheet-sized "game board" with two parallel lines cut by a transversal. Students will do DDT with two fingers while seated. Chair-dancing is encouraged.

I'm not sure how long it will take my students to master each level, so I've created a couple of alternate playlists in iTunes for Levels 2 and 3.

Also, at the end, we are going to have a dance-off!

Playlists for each level are as follows:

LEVEL 1 - The Honeyhive (I doubt we'll stay at Level 1, if we do it at all) 
LEVEL 2 - Herb Alpert, Mexican Shuffle
LEVEL 2 ALT - Raymond Scott, Powerhouse (middle section only, on a loop)
LEVEL 2 ALT 2 - Herb Alpert, Spanish Flea 
LEVEL 3 - Yakety Sax
LEVEL 3 ALT - The James Bond Theme (Original Version) 
LEVEL 4 - Mission Impossible
UPDATED 09/21/15:

And it was glorious. Here are two tiny video clips (one of the screen and one of the "dance"):



Saturday, September 19, 2015

Proportional Reasoning Capture Recapture with Goldfish Activity (Algebra 1)

I recently ran Julie's version of Capture Recapture with Goldfish activity with my Algebra 1 classes today. It was a huge hit!

This is a middle school topic, but proportional reasoning is so important it needs to be repeated every year in basic high school courses. Driving past the topics is not supposed to be the point! In HPL terms, I thought of this as an "activate prior knowledge" task that we tried to fit firmly into place with an engaging transfer task.

Julie's recommendation about showing the video is spot on: you absolutely MUST show the video. I broke it down as follows:

  • the first 1:30 to reveal the problem and the general idea (but without revealing the math or the solution)
We then discussed what was going on (activating prior knowledge about proportional reasoning) and together, we remembered how to do these problems.

Only after we had done our own work to rediscover the mathematics did I show the next 33 seconds of the video:

  • the next 33 seconds to reveal the mathematics needed to estimate
Together we wrote down all the figures and elements of the problem as Johnny Ball had revealed them so far. Then everybody in our class did their own work to figure out how many ping pong balls they estimated would be in his fish tank.

Once we had done that, then I did the final big reveal:

  • the next 13 seconds to reveal the exact number of balls in the tank

Once they had seen all of this, it was time for a transfer task.

I had students work in groups, recording their four trials on their data worksheets, taking an average of their estimates for each trial, and finally counting out their exact number of goldfish and writing a summary statement about how close (or far off) they had been. When they were done, they sent a representative to the whiteboard to add their group's data and best estimates to our table of whole-class results.

When we had everybody's data and estimates added to the table on the whiteboard, we discussed how accurate this method seemed and came up with other situations in which this method would be useful.

The hardest thing in Algebra 1 in my opinion is getting students to stop seeing skills and concepts as being discrete topics that you can mentally "put away" after the chapter test and to start seeing skills and concepts as new tools you want to "keep close at hand" in your mental tool belt.

So connecting proportional reasoning to tangible (and often edible) results in the real world is at least as important as the content of the lesson itself. This goes beyond merely "spiraling" back; it requires integrating into mathematical thinking and problem-solving from each moment forward.

Sunday, September 13, 2015

"How People Learn" and how people learn

How People Learn (HPL) is back in the blogs again, and for me, that is always a good thing. There is so much value, depth, and humanity in this slim, free book by the National Academies Press that any time anybody wants to talk about it at all, I say let's mark that as a win in the 'Wins' column.

There seems to be some misunderstanding, though, about exactly what HPL proposes an effective learning cycle ought to look like. Since in HPL, there is a place for everything, here is my 30,000-foot understanding and implementation of the four-stage process it advocatesI don't claim to be the definitive voice in any of this. I'm just taking this opportunity to document and share my practices in using their model because I believe that understanding this framework can go a long way toward helping teachers make good instructional decisions that can help their students to learn and thrive.

Specifically, HPL advocates:
STAGE 1 - a hands-on introductory task designed to uncover & organize prior knowledge. In this stage, collaborative activity provides an occasion for exploratory talk so that students can uncover and begin to organize their existing knowledge;
STAGE 2 - initial provision of a new expert model, with scaffolding & metacognitive practices woven together. The goal here is to help students bring their new ideas and knowledge into clearer focus so that they can reach the next level. Here again,  collaborative activity can provide a setting in which to externalize mental processes and to negotiate understanding, although often, this can be a good place to offer some direct instruction;
STAGE 3 - what HPL refers to as "'deliberate practice' with metacognitive self-monitoring." Here the idea is to use cooperative learning structures to create a place of practice in which learners can work within a clearly defined structure in which they can advance through the 3 stages of fluency (effortful -> relatively effortless -> automatic)
STAGE 4 - working through a transfer task (or tasks) to apply and extend their new knowledge in new and non-routine contexts. 
As with all good models, there is a lot of fluidity and variation in each stage, depending on how the teacher "reads" the learners in her classroom.  Here are some of my notes on each of the stages and how I have learned to look at each stage realistically and pragmatically:

A good discovery activity can be a powerful catalyst for learning  in Stage 1. But unfortunately, sometimes there just really isn't a great discovery activity that leads students captivatingly but inexorably to a blinding insight that will transform their learning forever.

Sometimes the best you've got is a mediocre discovery activity from a textbook that kinda sorta leads students in the general direction — but not without a lot of heavy-handed guidance. Or perhaps there is some other deficiency in what is available to you.

Like Gattegno, I believe that all learners have an energy "budget," and that means I have to make savvy and strategic decisions about how I'm going to ask my students to apply theirs. A boring or mediocre discovery activity can take just as much energy as a great one, but without the payoff of leaving students energized.

So sometimes I've learned I have to ask myself, is a discovery activity the best choice I can make here at Stage 1? Or do I have some other kind of introductory task I could use — such as a simulation, a story, a funny or interesting deleted scene, or some other kind of analogy — that will get my class into the learning episode faster and free up more of their energies to developing the necessary fluency that a rich and interesting transfer task may require?

To me, the most important thing that can happen in Stage 1 of a learning episode is that students come sharply to appreciate the Burning Question of this segment. Whenever possible, I really like for my students to arrive at a Burning Question through a collaborative discovery activity that they own because when they own it, they buy into it.

But realistically, this is simply not always possible with every single topic in the curriculum. So I have a range of strategies for Stage 1 that can get my students to a Burning Question even though there may be a gap in my pedagogical arsenal.

If the purpose of Stage 1 is to motivate students to ask a Burning Question, then the purpose of Stage 2 is to provisionally "pay off" the Burning Question — and to whet their appetite for knowing more. I say that my purpose here is to provisionally pay off the Burning Question because I believe a huge part of growing up as a learner is developing your own internal capacity for identifying questions and finding ways to pay them off and extend them.

So for me, this is where I "earn" the right to give my student a little bit of lecture, although when I work with them, I always call it "doing some notes" or "organizing our ideas" or "investigating ways in which others before us have thought about this problem." I say this not because I'm trying not to admit that I am lecturing (I am lecturing here) but I am also modeling note-taking and annotating practices that they will need when they arrive at a class where there is no other learning mode than lecture. No matter who you are and no matter where you study, at some point, somebody is going to lecture at you. If you are lucky (like I was at Princeton), those who lecture at you will consider it a high art form and will put great thought and care into their storytelling and argumentation modes.

Realistically, though, a lot of the lecture we encounter in our lives is not thrilling. But you need a certain degree of note-keeping and annotating skills that will enable you to survive those instructors and their inanimate lecturing practices so you can take what you need from their teaching and move on in your life.

So I use Stage 2 to also teach my students these note-taking/note-keeping/annotation survival skills as well as some metacognitive practices that will help them to get the greatest possible "bang" for their note-taking "buck."

As HPL clearly says, Stage 2 is about the "initial provision of an expert model." This is the place where we are sharing what students cannot find or develop on their own — or at least, what they cannot find or develop very efficiently given the time constraints of teaching and learning.

So please don't tell me there's no place for a transmission model in the HPL learning cycle. It's there, we all do it, and we all need to do it from time to time. Enough said. Let's move on.

With some new knowledge or ideas in hand, and having borrowed a more expert model from me as a tradeoff for accelerating the learning cycle, students need time to practice thinking these new thoughts, using the new model, and discovering what happens when they take it out for a spin. Deliberate practice with metacognitive self-monitoring is not the same thing as drill-and-kill. It's a form of experiential learning, like what a young child develops as they are integrating new vocabulary words. I've heard that a toddler needs to hear a new word used appropriately in context between 10 and 20 times before s/he can try it out for herself or himself. Mathematical ideas are no different. Students need to try and stumble, try and wobble, try and fall over, dust themselves off and try again until something takes hold in their unconscious. Nobody really knows what this secret crossover point is for every learner in every subject and every topic. So we provide a range of experiences for our students to help them find this crossover point for themselves.

Once students achieve some degree of "relatively effortless" fluency, they can dive into a transfer task.To me, an inspiring transfer task is more important than all of the mediocre discovery tasks in the world combined. An inspiring transfer task takes a learner seriously as a professional, and offers him or her an engaging, in-context opportunity to apply their new learning with all its glorious, messy, gravity-driven moving parts. One lightbulb moment from a transfer task — say, as Barbie is launched over a balcony railing, held aloft only by a series of looped rubber bands in answer to the question "How do we balance 'thrilling' and 'dangerous' to give her the greatest possible bungee jump that does not split her little plastic skull open?" — can last a lifetime. Being tasked to figure out experimentally and quantitatively whether or not Double Stuf Oreos do indeed contain double the "Stuf" as regular Oreos... or whether they are another marketing fraud being perpetrated on the Oreo-eating public can easily push a student over the fence into losing themselves in doing mathematics.

And frankly, to me, that is the whole point.

Wednesday, September 2, 2015

"Find What You Love; Do More of It"— #MTBoS Edition, a report from the field

While my friend @TrianglemanCSD Christopher Danielson is holding down the fort at the Minnesota State Fair and the Math On A Stick exhibit, I'm in my third full week of school avec students, and I realized I needed to peel off another layer of my persona and take his advice from his keynote address at #TMC15:
Find what you love; do more of it.
I love storytelling and stories. I come from a family of storytellers, where dinnertime was always a time of sharing stories.

I also love history — ancient history — and I know more about it than I usually give myself credit for.

So I did more of both of those things today. In my Algebra 1 class, no less.

It felt like a huge risk. But I decided to feel the fear and do it anyway.

So I told them what I have long known about the origins of algebra and equations. In its earliest uses, "algebra" means "balance." An equation is a metaphor for what everyone in the ancient world knew and understood with their own inherent sense-making and mathematical reasoning: just as a vendor at a market weighs out what a customer wants to buy — weighs it out with standards-based measures that are sanctioned by governments and universally accepted — so too is an equation a representation of these scales... and our goal is to balance out abstract or concrete quantities using that familiar structure from daily life.

I asked students what they knew about weighing and measuring and they told me honestly what their experiences have been at the farmer's markets all over our city.

We investigated what we knew about what happens when you place a known amount of weights on one side of a balance, and we imagined — and in some cases, acted out — what happens as you pour a continuous quantity of something onto the other side of the balance.

We talked about what it means to bring a scale into balance and we applied what we knew — the best we had — to the ideas at hand.

We talked about national and international standards bodies and about how even a perfect model degrades over time, which is why it needs to be monitored and occasionally replenished.

And so, by the time we got to the addition, subtraction, multiplication, and division properties of equality, we were already deep into our own connections with the metaphor and with the human history of algebra and with our own active and vivid imaginations.

By the end of class, nobody even complained about having to do the 2-2 homework. I am hoping that's because it was a little more deeply connected to their humanness than it ever had been before.

Saturday, August 29, 2015

New Geometry Unit 0 — Intro to Logic

Since, as @samjshah reminds me, we blog partly as an reflective archive for ourselves, I wanted to capture some of what I did in the new introductory logic unit I created for Geometry this year. I also wanted to capture some of who and what inspired me to do the things that worked out best!

I had no idea how prescient it would be to have an introductory unit that both adds tremendous rigor and depth while simultaneously being somewhat optional. These first two weeks of school have been somewhat chaotic as we discover who has been placed and scheduled correctly, who needs to be placed into a different section due to unavoidable schedule changes, corrections, or updates, how many new sections of certain courses we need, and so on. There were days when I felt like I didn't have the same students in any class two days in a row — even though some days this was more of a feeling than an actuality. So it was wonderful to have an adaptable "throughline" these two weeks — a river into which students might step, flow, or reenter without too much extra craziness. The unit was hard in the good way that students at my school really love — at the Zone of Proximal Development (ZPD), way up on a shelf just high enough above them that they have to stretch to reach it.

DAY 1 — Attacks and Counterattacks — What makes a mathematical definition?
We started with @samjshah and Brendan's Attacks and Counterattacks, which is now the recommended default starting unit for all Geometry students in our district. It was a great icebreaking activity, prompting students to activates their prior knowledge about what constitutes a definition. Plus it involved defining a narwhal, so how could that end badly, right? Table groups passed their definitions around the room, then used table-sized whiteboards to come up with a counterexample that broke the defining group's definitions. This involved both collaboration and presentation skills, as well as a good memory for definitional trivia. Did you know that the horn of the narwhal is ACTUALLY a tooth?

DAY 2 — Statements, Compound Statements, and Truth Tables
I also used this unit to set up classroom norms. Each day, when students come in, the "Welcome" slide is projected onto the board with introductory instructions and the Home Enjoyment (HE) assignment to copy down into your notebook.

We don't have bells at our school, so as an auditory cue, I am taking a page from the amazing @MrVaudrey and embedding a one-minute "welcome to class" music button that tells students this is a short, specific, and finite portion of our show and they know what they need to do. Thank you, Mr. Vaudrey! The theme music for Geometry (my 1st block class at 7:35 a.m.) is the music from Hawaii 5-0 (in honor of my principal and reminding me to ask myself, What Would @wahedahbug Do? with her brilliant mathematical classroom intro routines).

The instructions sometimes tell students to grab a handout from the handouts hanger but they always tell students to get out their HE and compare answers. I have realized that if I need to include homework review during a time of greatest primacy and recency, then I am going to make it count (thank you, @druinok and @pamjwilson!).

Next up was a dramatic table read of a deleted scene from the first Harry Potter movie. It's just terrible what ends up on the cutting room floor but, you know — Hollywood. "Harry Potter and the Logical Statement" was a SMASH hit. Students taught themselves the basics of statements, negations, equivalence, and truth tables and it beat the living crap out of giving them a boring lecture. After they were done, we summarized and organized what we had learned and I set up expectations for Home Enjoyment.

Day 3 —Advanced Equivalence & Rules of Replacement; Intro to Conditional Statements
The beginning of the truth table Olympics. Getting students to use what they know and extend it and — hooray — organize their work logically on the page was a huge win. Plus the kids liked learning something grown-up and hard. We did more practice set up homework review routines and expectations. Students are getting better about coming into class and getting started right away. Overview of the Professionalism score and expectation-setting.

Day 4 —Conditional Statements, Part II
Students discovered the Law of Non-Contradiction (Law of the Excluded Middle: A ⋁ ~A, but not both), and introduction to converse, inverse, and contrapositive using the basic conditional statement, "If I am Batman, then I am a superhero" (I'm looking at you, @mgolding). Whole lot of truth table whiteboarding going on.

Day 5 —Four Basic Laws of Inference
Modus ponens, modus tollens, simplification, disjunctive syllogism, and intro to proof. By this point are really getting into the technical language, to my great surprise. We start whiteboarding proofs and advanced truth tables.

Day 6 — Biconditionals & Definitions
We return full circle to where we started, with definitions, but now we define biconditional (iff) statements and how to prove them. Students start to groove on the idea that you have to prove both if A, then B and also if B, then A to prove a biconditional. There are lightbulb moments about the importance of counterexamples. Routines are starting to gel. Students are still transferring into the class and between/among sections/instructors, but other students help to indoctrinate them into our emerging culture.

Day 7 — Intensive Practice Day — Truth Table Practice aka Whiteboarding Madness
I give each table a sheet of truth tables to build and I use this activity as a Participation Quiz to further solidify our norms. Many groups start passing the marker around to ensure equitable participation. Everybody does splendidly on the Participation Quiz.

Day 8— Assessment
Home Enjoyment Packet #1 is due to be turned in. Students take the Unit 0 Logic test. It is a bear, but my students are "scared but prepared." Some students need more time so they come in at 7:15 a.m., during 3rd, 4th, 5th, or 6th block to finish. Their stillness and concentration is impressive. Algebra 1 students are doing their own crazy stuff all around them, but they persist and persevere. I am over the moon about them. They are proud of themselves for the advanced logic they have learned.

Day 9— Chapter 1 Launch: The Elements of Geometry — the poetry of primitives
We explore undefined terms, learn a little math history, and do some Think-Pair-Share. Next year I want to have another table reading/deleted scene activity for all this stuff instead of a boring lecture. But that is OK. Our routines are solid and we are moving forward.

Onward to constructions on Monday.

Now I have to score 73 logic tests (and 73 HE packets, but only for completion), but it was totally worth it. My Geometry classes are off to a great start, and we have a solid foundation to build from, even if my sections have 36 kids each.

Saturday, August 22, 2015

TMC15 reflection: The Story Of TMC or, How We Didn't Get Lucky

We had our first week with students this week and boy, am I tired.

But all week long, I feel like I've been carried along on the current of good energy I have forged over the years with my TMC (Twitter Math Camp) math teacher tribe.

I was thinking about how other people keep telling me, Oh wow, you're so lucky you've got that.

And I finally realized I've been wanting to say, "No — we didn't get lucky."

A little over four years ago, a bunch of us who had met on Twitter and blogs decided we wanted to get together in real life. In December of 2011, over winter break, there was what is now known as The Great Facebook Friending of 2011. One night during a rampage of funny, crazy, meaningful tweeting among math teacher tweeps, we made the decision to "Facebook-friend" each other.

At the time, that felt like a HUGE risk — letting other people into our real, personal lives.

I was worried that the next morning I would wake up and discover that it had all been an enormous mistake and I would need to go into internet witness protection to get away from these crazies. I was worried that I was going to have such a hangover.

But no. I discovered that these really WERE the people I wanted to be connected with. And other people did too.

So even though Julie Reulbach still wanted us to go on a cruise together, we all decided it would be safer — and saner — to meet on land somewhere. The Mary Institute and Country Day School in St. Louis was gracious enough to offer us a free space to have our math teacher jamboree, and we all traveled from remote parts of North America on our own dimes to get there.

And as I like to remind people who say how lucky we were this year to have TMC at Harvey Mudd College in Southern California — remember that at the first TMC in St. Louis, I was the only person from California who showed up.

We didn't get lucky. We took small, incremental risks with our teaching and with our professional development until we felt safe enough and ready enough to form something larger.

And as the great psychoanalyst and cantadora Clarissa Pinkola Estes has said, when you step forward and truly embrace your whole life with your whole life, other like-minded people will "mysteriously show up, announcing that this is exactly what they have been looking for all along."

In other words, there are rewards for courage.

So my biggest TMC15 reflection is a reminder to myself that we did NOT in any way just "get lucky" with TMC. We stepped forward and showed up in our professional development lives — over and over and over. We stepped over the negative chatter of people all around us saying "there's no such thing as good PD" and we pushed past people who asked negative questions like "Why would you want to use part of your summer vacation time to travel to ______ [fill in the blank with St. Louis/Philadelphia/Jenks, OK/Pasadena] in the summer for professional development?" and we ignored the negativity of anybody in our home districts dumping on "Common Core math" or "all that fancy-schmancy group work nonsense."

We took a deep reflective breath and said a holy "yes" to being deliberate about our teaching practice and taking the risk of investing our whole selves into it. And THAT, as I constantly remind my students, is the essence of "luck."

Wednesday, August 19, 2015

DELETED SCENE: Harry Potter and the Logical Statement — GEO (statements, compound statements, and truth tables)

I'm absolutely slammed for time, but I wanted to share this script I wrote for my Geo students, which was a surprise hit.

I'm taking a page this year from the patented Sam Shah "Here kids, teach yourselves this stuff" pedagogical method, which really deserves a lot more credit than it gets.

I had students act out this "script" which is, of course, from a confidential deleted scene from the first Harry Potter movie. The studio insisted I return all copies after the class. Students delighted in finding the many "problems" with the script and we laughed about all the many reasons why this scene was obviously deleted.

When you release students to launch the task, you want to slam a ruler on the table and yell, "ACTION!"

Students had a lot of fun and got their intro to symbolic logic. Mission accomplished.


The file is available on the Math Teacher Wiki:

Deleted Scene: Harry Potter and the Logical Statement